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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution</dfn>:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
p(x)=x \frac{Q(x)}{P(x)}=x \frac{x}{x^2}=1,\quad q(x)=x^2 \frac{R(x)}{P(x)}=x^2 \frac{x^2-(1/2)^2}{x^2}=-\frac{1}{4}+x^2.
\end{equation*}
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<p class="continuation">Since both <span class="process-math">\(x Q(x)/P(x)\)</span> and <span class="process-math">\(x^2 R(x)/P(x)\)</span> are analytic at <span class="process-math">\(x=0\text{,}\)</span> <span class="process-math">\(x=0\)</span> is a regular singular point.</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
p_0=\lim \limits_{x \to 0} x \frac{Q(x)}{P(x)}=1,\quad q_0=\lim \limits_{x \to 0} x^2 \frac{R(x)}{P(x)}=-\frac{1}{4}.
\end{equation*}
</div>
<p class="continuation">The indicial equation is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
r^2+(p_0-1) r+q_0=0 \to r^2-r+r-\frac{1}{4}=0 \to r^2-\frac{1}{4}=0 \to r_1=\frac{1}{2},\quad r_2=-\frac{1}{2}.
\end{equation*}
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<span class="incontext"><a href="sec5_5.html#p-235" class="internal">in-context</a></span>
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